[Paper Review] Test Sets for Integer Programs with Z-Convex Objective

Raymond Hemmecke|ArXiv.org|Sep 8, 2003

Advanced Optimization Algorithms Research8 references6 citations

TL;DR

This paper extends test set methods from integer linear programming to integer programs with ℤ-convex objective functions by constructing finite, computable test sets using Graver bases. It proves that the projected Graver basis of an extended matrix provides improving directions for any non-optimal solution, enabling augmentation-based algorithms for broader classes of integer programs including quadratic and non-smooth problems.

ABSTRACT

In this paper we extend test set based augmentation methods for integer linear programs to programs with more general convex objective functions. We show existence and computability of finite test sets for these wider problem classes by providing an explicit relationship to Graver bases. One candidate where this new approach may turn out fruitful is the Quadratic Assignment Problem.

Motivation & Objective

• To generalize test set methods beyond linear integer programs to handle ℤ-convex objective functions.
• To establish the existence and computability of finite test sets for integer programs with ℤ-convex objectives.
• To provide a constructive method for generating improving directions using Graver bases of an extended problem matrix.
• To demonstrate applicability to challenging problems like the Quadratic Assignment Problem.
• To unify the treatment of non-linear, non-smooth, and convex integer programs under a single algorithmic framework.

Proposed method

• Constructs an extended integer program by introducing binary variables to model the piecewise-linear structure of ℤ-convex functions.
• Uses the Graver basis of the extended problem matrix to generate improving directions.
• Projects the Graver basis elements back to the original variable space via a projection operator φₙ.
• Defines the test set H_CIP(A,C) as the projection of the Graver basis of the matrix (A | 0; C | Iₛ).
• Leverages the property that ℤ-convex functions have increasing differences, enabling discrete convex analysis.
• Applies a transformation to reduce any ℤ-convex function with minimum at α to one with minimum at 0.

Experimental results

Research questions

• RQ1 Can finite test sets be constructed for integer programs with ℤ-convex objective functions?
• RQ2 Is there a systematic way to generate improving directions for non-linear integer programs using Graver bases?
• RQ3 Can the augmentation method be extended beyond linear objectives while preserving finite convergence?
• RQ4 How does the structure of ℤ-convex functions relate to the existence of universal test sets?
• RQ5 What is the relationship between the Graver basis of an extended formulation and the test set for the original problem?

Key findings

• A finite test set H_CIP(A,C) exists and is computable for any integer program with ℤ-convex objective function.
• The test set H_CIP(A,C) is derived as the projection of the Graver basis of the extended matrix (A | 0; C | Iₛ).
• Any non-optimal feasible solution can be improved by a vector in H_CIP(A,C), ensuring convergence to an optimal solution.
• The method generalizes to non-smooth and non-quadratic ℤ-convex functions, such as absolute values and exponential functions.
• The test set includes the original Graver basis of A, ensuring compatibility with linear integer programs.
• The approach enables algorithmic solution of problems like the Quadratic Assignment Problem via augmentation.